Toward computability of trace distance discord

It is known that a reliable geometric quantifier of discord-like correlations can be built by employing the so-called trace distance. This is used to measure how far the state under investigation is from the closest"classical-quantum"one. To date, the explicit calculation of this indicator for two qubits was accomplished only for states such that the reduced density matrix of the measured party is maximally mixed, a class that includes Bell-diagonal states. Here, we first reduce the required optimization for a general two-qubit state to the minimization of an explicit two-variable function. Using this framework, we show next that the minimum can be analytically worked out in a number of relevant cases including quantum-classical and X states. This provides an explicit and compact expression for the trace distance discord of an arbitrary state belonging to either of these important classes of density matrices.


Introduction
The issue that the quantum correlations (QCs) of a composite state are not entirely captured by entanglement (as formerly believed) has recently emerged as a topical subject calling for the introduction of new paradigms. Despite early evidence of this problem was provided over a decade ago [2] an impressive burst of attention to this matter has developed only in the last few years [1] as witnessed, in particular, by very recent experimental works (see e.g. Refs. [3,4]). In this paper, we focus on those correlations that are associated to the notion of quantum discord [2]. Following the introduction of this concept, a variety of different measures of QCs have been put forward (see Ref. [1] for a comprehensive review). A major reason behind such a proliferation of QCs indicators stems from the typical difficulty in defining a reliable measure that is easily computable. No general closed formula of quantum discord, for instance, is known (with strong indications that this is an unsolvable problem [6]) even for a pair of two-dimensional systems or "qubits" [5], namely the simplest composite quantum system. Unfortunately, the demand for computability typically comes at the cost of ending up with quantities that fail to be bona fide measures. In this respect, the most paradigmatic instance is embodied by the so called geometric discord (GD) [7], which while being effortlessly computable (and in some cases able to provide useful information) may entail unphysical predictions. It can indeed grow under local operations on the unmeasured party [8], an effect which a physically reliable (bona fide) indicator (e.g. quantum discord) is required not to exhibit. Following an approach frequently adopted for other QCs measures, the one-sided GD is defined as the distance between the state under study and the set of classicalquantum states. The latter class features zero quantum discord with respect to the measured party, say subsystem A, which entails the existence of at least one set of local projective measurements on A leaving the state unperturbed [2,9]). While the above definition in terms of a distance is clear and intuitive, it requires the use of a metric in the Hilbert space. The GD employes the Hilbert-Schmidt distance, which is defined in terms of the Schatten 2-norm. Such a distance is well-known not to fulfil the property of being contractive under tracepreserving quantum channels [10,11], which is indeed the reason behind the aforementioned drawback of GD [12]. This naturally leads to a redefinition of the GD in terms of a metric that obeys the contractivity property. One such metric is the trace distance [5,13], which employs the Schatten 1-norm (or trace norm for brevity). In the remainder of this paper, we refer to the QCs geometric measure resulting from this specific choice as trace distance discord (TDD).
While investigations are still in the early stages [14,15,16,17,18], TDD appears to enjoy attractive features, which makes it a physically meaningful measure. Besides the discussed contractivity property, the trace distance is invariant under unitary transformations. More importantly, it is in one-to-one correspondence with one-shot state distinguishability [19], i.e., the maximum probability to distinguish between two states through a single measurement. This operational interpretation provides evidence that the trace distance works as an accurate "meter" in the space of quantum states which, importantly, has a clear physical meaning. Another appealing advantage of TDD lies in its connection with entanglement. Recently, indeed, it was suggested to define the full amount of discord-like correlations in a system S as the minimum entanglement between S and the measurement apparatus created in a local measurement (see [20,21,22] and references therein). This way, a given entanglement measure [23] identifies a corresponding QCs indicator. Remarkably, it turns out that the latter always exceeds the entanglement between the subparts of S when this is quantified via the same entanglement measure. This rigorously formalizes the idea that a composite state can feature QCs that cannot be ascribed to entanglement. In this framework, it can be shown [16] that the entanglement counterpart of TDD is negativity [24], the latter being a well-knownin general easily computable -entanglement monotone [25].
In spite of all such interesting features, the easiness of computation of TDD in actual problems is yet to be assessed. To date, the only class of states for which a closed analytical expression has been worked out are the Bell-diagonal (BD) two-qubit states, or more generally states that appear maximally mixed to the measured party [16,17]. While the proof of this formula is non-trivial [16], this does not clarify whether or not, besides its reliability, TDD brings about computability advantages as well. Owing to the high symmetry and reduced number of parameters of BD states, indeed, most if not all of the bona fide QCs measures proposed so far can be analytically calculated for this specific class [27].
In this paper, we take a step forward and set up the problem of the actual computation of two-qubit TDD on a new basis. We first develop a theoretical framework that reduces this task to the equivalent minimization of a two-variable explicit function, which parametrically depends on the Bloch vectors of the marginals and the singular values of the correlation matrix. Next, after re-deriving the value of TDD for a class of density matrices that includes BD states, we discuss two further relevant cases in which the minimization problem can be analytically solved. One is the case where the correlation matrix has one non-zero singular eigenvalue, a subset of which is given by the quantum-classical states (unlike classicalquantum states these feature non-classical correlations with respect to party A). The other case is given by the family of X states [26], which include BD states as special cases. While these are arguably among the most studied classes of! two-qubit density matrices [1], the calculation of their QCs through bona fide measures is in general a demanding task. To the best of our knowledge, in particular, no closed expression for an arbitrary quantum-classical state is known to date with the exception of Ref. [28] where however an ad hoc measure exclusively devised for this specific class of states was presented. In the general case, indeed, one such state depends on four independent parameters and, moreover, features quite low symmetry. In Refs. [28] and [29], for instance, closed expressions for a fidelity-based measure [30] and the quantum discord, respectively, could be worked out only for high-symmetry twoparameter subsets of this family.
Even more involved is the calculation of QCs in the case of X states, a class which depends on five independent parameters. Regarding quantum discord, an algorithm has been put forward by Ali et al. [31]. Later, however, some counterexamples of X states for which such algorithm fails were highlighted [32] (see also Ref. [1]).
The present paper is organized as follows. In Section 2, we present our method for tackling and simplifying the calculation of TDD for an arbitrary two-qubit state. This is demonstrably reduced to the minimization of an explicit two-variable function. In Section 3, we apply the theory to the case of Bell states and that of density matrices having correlation matrix with uniform spectrum. In Section 4, we show that the minimum can be analytically found in a closed form whenever the correlation matrix of the composite state features only one non-zero singular value. As an application of this finding, in Subsection 4.1 we compute the TDD of the most general quantum-classical state. As a further case where the minimization in Section 2 can be performed explicitly, in Section 5 we tackle the important class of X states and work out the TDD for an arbitrary element of this. In Section 6, we illustrate an application of our findings to a paradigmatic physical problem (propagation of QCs across a spin chain), where the analytical calculation of quantum discord [2], although possible, results in uninformative formulas. We show that, while the time behavior of TDD exhibits the same qualitative features as the quantum discord, its analytical expression is quite simple. We finally draw our conclusions in Section 7. A few technical details are presented in the Appendix.

One-sided TDD for two-qubit states: general case
The one-sided TDD D (→) (ρ AB ) from A to B of a bipartite quantum state ρ AB is defined as the minimal (trace norm) distance between such state and the set CQ of classical-quantum density matrices which exhibit zero quantum discord with respect to local measurements on A, i.e. states which admit an unravelling of the form with |α j A being orthonormal vectors of A and B ( j) being positive (non necessarily normalized) operators of B. Specifically, if Θ 1 = Tr[ √ Θ † Θ] denotes the trace norm (or Schatten 1-norm) of a generic operator Θ then the 1/2 factor ensuring that D (→) (ρ AB ) takes values between 0 and 1 [analogous definition applies for the one-sided TDD from B to A, D (←) (ρ AB )]. The quantity in Eq. (2) fulfills several requirements which make it fit for describing non-classical correlations of the discord type [16]. In particular, from the properties of the trace distance [5] it follows that D (→) (ρ AB ) [33] i) is zero if and only if ρ AB is one of the classical-quantum density matrices (1); ii) is invariant under the action of an arbitrary unitary operation U A ⊗ V B that acts locally on A and B, i.e.
iii) is monotonically decreasing under completely positive and trace preserving (CPT) maps on B; iv) is an entanglement monotone when ρ AB is pure.
Furthermore, in the special case in which A is a qubit Eq. (2) can be expressed as [16] D (→) (ρ AB ) = 1 2 min where now the minimization is performed with respect to all possible completely depolarizing channel Π A on A associated with projective measurements over an orthonormal basis, i.e.
with P A ≡ |Ψ A Ψ| and Q A = I A − P A being rank-one projectors (|Ψ is a generic one-qubit pure state).
In what follows, we will focus on the case where both A and B are qubits. Accordingly, we parametrize the state ρ AB in terms of the Pauli matrices where is the Bloch vector corresponding to the reduced density matrix ρ A(B) describing the state of A(B), while Γ is the 3×3 real correlation matrix given by Similarly, without loss of generality, we express the orthogonal projectors P A and Q A of Eq. (5) as withê being the 3-dimensional (real) unit vector associated with the pure state |Ψ A in the Bloch sphere. Using this and observing that Π A (I A ) = I A , and Π A ( υ · σ A ) = (ê · υ) (ê · σ A ), Eq. (4) can be arranged as where the minimization is performed over the unit vectorê and M(ê) is a 4 × 4 matrix which admits the representation Here,x i is the ith Cartesian unit vector and e i =x i ·ê the ith component ofê (note that σ Ai =x i · σ A ). The second term in Eq. (11) can be further simplified by transforming Γ into a diagonal form via its singular value decomposition [34]. More precisely, exploiting the fact that Γ is real we can express it as where O and Ω are real orthogonal matrices of SO(3) while {γ i } are real (not necessarily nonnegative) quantities whose moduli correspond to the singular eigenvalues of Γ [35]. We can then define the two set of vectorŝ for k = 1, 2, 3. As O, Ω ∈ SO(3), by construction {ŵ k } is an orthonormal (right-hand oriented) set of real vectors and so is {υ k } (each is indeed a rotation of the Cartesian unit vectors {x j }).
Using the above, we can arrange Eq. (11) as where for compactness of notation we introduced the vectors to represent the orthogonal component of x A andŵ k with respect toê. Note that {υ k · σ B } describes the transformed set of Pauli matrices under a local rotation on B. This set clearly fulfills all the properties of Pauli matrices as well. One can therefore redefine the B's Pauli matrices as {υ k · σ B } → σ Bk , which amounts to applying a local unitary on B. Let then M (ê) be the transformed operator obtained from M(ê) under such rotation, i.e.
Since the trace norm is invariant under any local unitary we have (11) associated to the state ρ AB obtained from ρ AB via a local unitary rotation associated to the transformation {υ k · σ B } → σ Bk ]. The trace norm of M (ê) can now computed by diagonalizing the operator M (ê) † M (ê). For this purpose, we recall that given two arbitrary vectors { x, y }, the Pauli matrices fulfil the following commutation and anti-commutation relations as well as the identities σ A1 σ A2 = iσ A3 , σ A2 σ A1 = −iσ A3 and the analogous identities obtained through cyclic permutations (in the above expression "∧" indicates the cross product). Using these, we straightforwardly end up with where x A⊥ = | x A⊥ | (throughout, x = | x | for any vector x), χ is a tridimensional real vector of components while Q is a positive quantity defined as and finally ∆ is the operator This expression can be simplified by observing that since the w k⊥ 's are vectors orthogonal toê [see Eq. (15)] their mutual cross products must be collinear with the latter. Indeed, introducing the spherical coordinates {θ, φ} which specifyê in the reference frame defined by {ŵ k }, we have Substituting these identities in Eq. (23), the operator ∆ can remarkably be arranged in terms of a simple tensor product as where g is the vector g = (γ 2 γ 3 sin θ cos φ, γ 3 γ 1 sin θ sin φ, γ 1 γ 2 cos θ) , which is orthogonal to χ [36]. Next, observe that the operatorê · σ A of Eq. (25) is Hermitian with eigenvalues 1 and −1. Therefore, if {|0 A , |1 A } are its eigenvectors we can writê e · σ A = |0 A 0| − |1 A 1|. Plugging this and I A = |0 A 0| + |1 A 1| into Eq. (20) this can be arranged as which can now be put in diagonal form. Indeed, due to the aforementioned spectrum of x · σ, it has eigenvalues λ = Q+ x 2 A⊥ ±2 χ 2 +g 2 , each twofold degenerate [36]. Therefore, through Eq. (17) we end up with where x Aê θ φ Figure 1. (Color online) Schematics of the minimization procedure for calculating the trace distance discord D (→) (ρ AB ) of a two-qubit state ρ AB . The reference frame in which this is carried out is defined by the orthonormal set of three vectors {ŵ k }, where eachŵ k is associated with a real singular eigenvalue γ k of the correlation matrix [see Eqs. 8, 12 and 13]. This frame identifies a representation for the local Bloch vector x A [defined in Eq. 7]. All these quantities are drawn using solid black lines to highlight that, for a given density matrix ρ AB , they are fixed. Instead, the unit vectorê (red line) represents the direction along which a projective measurement on A is performed. In the optimization procedure,ê is varied until function h in Eq. (31) reaches its global minimum according to Eq. (32).
Note that Q, x A⊥ , χ and g are all functions ofê [cf. Eqs. (21), (22) and (26)]. As M(ê) 1 is a positive-definite function, finding its minimum is equivalent to searching for the minimum of its square M(ê) 2 1 . Thereby where the function h(ê) is defined as In conclusion, in the light of Eqs. (10), (27) and (30) We have thus expressed our trace-norm-based measure of QCs of an arbitrary state ρ AB as the minimum of an explicit function of the two angles {θ, φ} (0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π). Equation (32) is the first main finding of this paper. For clarity, all quantities involved in the minimization problem under investigation are pictorially represented in Fig. 2.

Bell diagonal states and states with homogeneous singular values
The optimization in Eq. (32) simplifies when the state possesses certain symmetries. In particular, by ordering the singular eigenvalues of Γ as (this convention is adopted only in the present section) one can show that at least for two classes of states ρ AB , which we label as 'A' and 'B', respectively. These are defined as We develop the proof in the following two subsections.

Bell diagonal states
States of class A, which include Bell diagonal states, are characterized by the property that the reduced density matrix of subsystem A is maximally mixed. For these, Eq. (32) was proven in Refs. [16,17] using an independent approach. Here, we present an alternative (possibly simpler) derivation based on Eq. (32). We point out that these states form a special subset of X states, which we will study in full detail in Section 5. Here, our goal is indeed to present a straightforward application of our method for calculating the TDD developed in the previous Section.

States with homogeneous |γ k |'s
Class B (see definition given above) includes, for instance, mixtures of the form ρ AB = Bloch vector of A : therefore according to Eq. (34) this state has a value for TDD given by To derive Eq. (34), we introduce the diagonal matrix T = diag(t 11 , t 22 , t 33 ) formed by the coefficients t 11 , t 22 , t 33 defined by the identities [it is clear from (35) that t j j can only take values ±1]. Under this condition, from Eqs. (21), (22) and (26) it then follows where ξ takes value either 1 or −1 depending on the explicit form of the mapping (39). Replacing this into Eqs. (28), (29) and (31) we end up with

Correlation matrix with a single non-zero singular eigenvalue
This class of states is important since quantum-classical states fall within it, as we show later. It is defined by [see Eq. (12)] γ 2 = γ 3 = 0 while γ 1 = γ and x A are arbitrary (the only constraint is that the resulting ρ AB must be a properly defined density matrix). We show below that the TDD of one such state is given by where γ 1 = |γ 1 |ŵ 1 ,ŵ 1 being the first element of the set {ŵ k } defined in Eq. (13). Eq. (43) is another main finding of this work.
To begin with, we observe that due to γ 2 = γ 3 = 0 we are free to choose the direction of the Cartesian axesŵ 2 andŵ 3 (ŵ 2 ⊥ŵ 3 ) on the plane orthogonal toŵ 1 . We thus takeŵ 2 as lying on the plane formed byŵ 1 and x A . Hence we can write x A =x A1ŵ1 +x A2ŵ2 , wherex A1 andx A2 are the components of x A in reference frame defined by {ŵ k }. Accordingly, with α being the angle between x A andŵ 1 while x A = x 2 A1 +x 2 A2 . With the help of Eqs. (21), (22) and (26), in the present case a and b [cf. Eqs. (28) and (29)] read whereẽ 1 =ê ·ŵ 1 . Observe then that we can write It turns out that both a+ √ b and a− √ b decrease when the component ofê on the plane formed byŵ 1 and x A , i.e., theŵ 1 −ŵ 2 plane, grows. To see this, we decomposeê asê = ε + ε ⊥ , where ε =ẽ 1ŵ1 +ẽ 2ŵ2 is the component ofê on theŵ 1 −ŵ 2 plane, while ε ⊥ =ẽ 3ŵ3 the one orthogonal to it. With these definitions, in Eq. (46) we can evidently replaceê with ε (we remind thatx A3 = 0). Now, it should be evident that the last term of Eq. (46) can be written as is a function of φ (i.e., the azimuthal angle ofê) and the aforementioned α. Clearly, for given φ the minimum of a ± √ b is achieved when | ε| is maximum, i.e., for ε ≡ê or equivalently θ = π/2. Thus, due to Eq. (27), in Eq. (32) we can safely restrict the minimization overê = (θ, φ) to the set e = (π/2, φ). To summarize, we need to calculate Through few straightforward steps (see Appendix A), M(ê) 1 can be arranged as (we henceforth omit to specificy θ = π/2) Exploiting the positiveness of M(ê) 1 and the identity (|y+z|+|y−z|) 2 = 4 max{y 2 , z 2 }, where y and z are any two real numbers, Eq. (48) can be converted into Replacing ||M(ê)|| so obtained into Eq. (10) we can then express the one-sided TDD of our state ρ AB in terms of the following min-max problem, An analytic solution is obtained by observing that the φ-dependent functions f 1 (φ) = | sin β sin(φ − α)| and f 2 (φ) = | cos β sin(φ)| have the same period π and that in the domain φ ∈ [0, π] exhibit the two crossing points φ c+ and φ c− given by By construction, the function Eq. (49) reaches is minimum either in φ c+ or in φ c− . Therefore, where the latter identity have been obtained through simple algebraic manipulations. To arrange this formula in a form independent of the reference frame, we make use of Eqs. (44) and (50). This finally yields Eq. (43).

Quantum-classical states
The result of the previous section can be exploited to provide an analytical closed formula of D (→) (ρ AB ) for the well-known class of quantum-classical states. One such state reads where ρ 0(1) is an arbitrary single-qubit state with associated Bloch vector s 0(1) , i.e, ρ 0(1) = I+ s 0(1) · σ /2. The state in Eq. (54) represents a paradigmatic example of a separable state which is still able to feature A → B quantum correlations. On the other hand, note that the quantum discord in the opposite direction, B → A, is zero by construction. One can assume without loss of generality that s 0 = (0, 0, s 0 ) and s 1 = (s 1 sin ϕ, 0, s 1 cos ϕ) with 0 ≤ ϕ ≤ π, i.e., the Z-axis of the Bloch sphere is taken along the direction of s 0 while the Y-axis lies orthogonal to the plane containing both s 0 and s 1 . Vector x A and matrix Γ are calculated as Γ has only one singular eigenvalue since its singular value decomposition yields γ 2 = γ 3 = 0 and |γ 1 | = γ = p 2 s 2 0 +(p−1)s 1 (p−1)s 1 +2ps 0 cos ϕ .
This, together with Eq. (55), yields the identities Replacing these into Eq. (43), we end up with which represents the TDD of the most general quantum-classical state [Eq. (54)]. This formula has a very clear interpretation in terms of the lengths of the local Bloch vectors on A, s 0 , s 1 , the angle between them ϕ and the statistical weights p, 1 − p. One can see that the maximum value of D (→) is 1/4 and is obtained for s 0 = s 1 = 1, p = 1/2 and ϕ = π/2: this corresponds to picking on system A two pure states with orthogonal Bloch vectors, that is, two vectors belonging to mutually unbiased bases. Indeed, for these parameters, Eq. (54) reduces to , which is a paradigmatic example of separable but quantum-correlated state. The qualitative behavior of D for s 0 = s 1 and p = 1/2 is fully in line with that of the quantum discord [29] and for s 0 = s 1 = 1 with that of the fidelity-based measure analyzed in Ref. [28].
In what follows, we will prove that the TDD of state Eq. (63) is given by showing that for the X-states the discord is only a function of the following three parameters: |γ 1 |, |γ 3 | and γ 2 2 + x 2 A3 . To begin with, Eqs. (28) and (29) imply that the (θ, φ)-dependent functions a and b entering Eq. (31) [recall that (θ, φ) specifyê] depend only on µ ≡ sin 2 θ and ν ≡ sin 2 φ as where {a i } and {b i } are the following linear functions of ν Clearly, a(µ, ν) and b(µ, ν) are defined in the square S defined by S ≡ {µ, ν : 0 ≤ µ ≤ 1, 0 ≤ ν ≤ 1} [and so is h = a+ As witnessed by the denominator of this equation, we observe that function h is in general non-differentiable at points such that a 2 = b, owing to the square root √ a 2 −b appearing in its definition, Eq. (31). One then has to investigate these points carefully, as they may potentially yield extremal values of h that would not be found by simply imposing ∂ µ h = ∂ ν h = 0.
As a key step in our reasoning, we first demonstrate that a minimum of h cannot occur in the interior of S. Afterward, we minimize function h on the boundary of S, which will eventually lead to formula (65).
Let us now address singular points, i.e., those at which h is non-differentiable and hence minimization criteria based on partial derivatives do not apply. These points (see above discussion) are the zeros of the function f = a 2 −b. Our aim is proving that even such points, if existing, lie on the boundary of S. Firstly, note that f ≥ 0 (we recall that a 2 ≥ b always holds, see Section II). This means that a zero of f is also a minimum point for f . From Eqs. (67)-(70) it is evident that f (µ, ν) is analytic throughout the real plane. Then a necessary condition for this function to take a minimum is ∂ µ f = ∂ ν f = 0. It is easy to check that ∂ ν f is a simple seconddegree polynomial in µ, with zeros µ s1 = 0 and µ s2 = (γ 2 1 +γ 2 2 −2γ 2 3 +2x 2 A3 )/[(γ 2 1 −γ 2 3 +x 2 A3 )+(γ 2 2 −γ 2 1 )ν]. The former solution clearly cannot correspond to stationary points of f -in particular zeros of f , i.e., singular points of h -that lie in the interior of S (as anticipated, a zero of f is also a minimum and thus one of its stationary points). On the other hand, by plugging We have already shown (see above) that in neither of these two cases h can admit minima in the interior of S.

Minima on the boundary of S
The findings of the previous subsection show that we can restrict the search for the minimum of h to the boundary of S. The possible values of h on the square edges corresponding to µ = 0, µ = 1, ν = 0 and ν = 1 are, respectively, given by From Eq. (72) it trivially follows that the minimum of h on edge µ = 0 is given by min h µ=0 = 2γ 2 1 . In the next three dedicated paragraphs, we minimize h on edges µ = 1 and ν = 0, 1, respectively.
As already anticipated, for Bell-diagonal states (see Section 3), Eq. (65) yields the result of Section 3.1 as shown in detail in Appendix B.

Application: propagation of QCs across a spin chain
In this Section, we present an illustrative application of our findings to a concrete problem of QCs dynamics. The problem was investigated in Ref. [38] and regards the propagation dynamics of QCs along a spin chain. Specifically, consider a chain of N qubits each labeled by index i = 1, .., N with an associated Hamiltonian Such XX model is routinely used to investigate quantum state transfer [39]. An additional qubit, disconnected from the chain and denoted by i = 0, initially shares QCs with the first qubit of the chain corresponding to i = 1 (with each of the remaining qubits initially prepared in state |0 ). The problem consists in studying how the bipartite QCs between qubits 0 and r with r = 1, ..., N evolve in time. If r = N, in particular, one can regard this process as the end-to-end propagation of QCs across the spin chain. In Ref. [38], the authors found a number of interesting properties, especially in comparison with the corresponding entanglement propagation. To carry out their analysis, they used the quantum discord D (→) Z [2]. For the specific two-qubit states involved in such dynamics, D (→) Z can be calculated analytically. Yet, this circumstance does not yield any advantage in practice since the resulting formulas are lengthy and uninformative, as pointed out by the authors themselves [38]. We next provide evidence that, if instead of D (→) Z , one uses the TDD D (→) then simple and informative formulas arise.
It is easily demonstrated [38] that if ρ 10 = (I 10 +σ 11 σ 01 )/4 is the initial state of qubits 1 and 0 then at time t the state of N and 0 reads where f (t) is the single-excitation transition amplitude given by Therefore, f fully specifies the output state (89) and thus any corresponding QCs measure. Fig. 2(a) shows, in particular, the behavior of | f (t)| and D (→) Z [ f ((t)] for N = 3, which fully reproduces the results in Ref. [38] (in absence of magnetic field). The quantum discord is evidently a non-monotonic function of | f |, which vanishes for | f | = 0, 1 exhibiting a single maximum at an intermediate value of | f |. There appears to be no straightforward way to prove this behavior since, as anticipated, function D (→) Z ( f ) has a complicated analytical form. Let us now calculate D (→) ( f ). State Eq. (89) is an X state, hence our techniques of Section 5 can be applied to calculate the corresponding TDD [40]. Using the notation of Section 5 and observing that off-diagonal entries in Eq. (89) can be replaced by their moduli (up to local unitaries that do not affect TDD) we find γ 1 = | f (t)|, γ 2 = γ 3 = 0 and x A3 = 1−| f (t)| 2 . Substituting these in Eq.66 then yields the compact expression which is plotted in Fig. 2(b). Once f is expressed as a function of time with the help of Eq. (90) we obtain the non-monotonic time behavior of D (→) displayed in Fig. 2(a). This exhibits the same qualitative features as D (→) Z (t), which shows that TDD has a predictive power analogous to the quantum discord. Unlike the latter, though, acquiring analytical insight is now straightforward. Indeed, it is immediate to see from Eq. (91) that D (→) vanishes for | f | = 0, 1. Moreover, the equation dD (→) /d| f | = 0 (which is easily seen to be equivalent to an effective 3rd-degree equation) admits only one root in the range [0, 1] given by | f | M 1/ ]. This paradigmatic instance illustrates the effectiveness of our findings as a tool to acquire readable and reliable informations on QCs in a concrete physical problem.

Conclusions
In this paper, we have addressed the issue of the computability of TDD, one of the most reliable and advantageous QCs indicators. By introducing a new method for tackling and simplifying the minimization required for its calculation in the two-qubit case, we have demonstrated that this can be reduced to the search for the minimum of an explicit twovariable function. Then, we have shown that this can be analytically found in a closed form for some relevant classes of states, which encompass arbitrary quantum-classical and X states. The latter includes as a special subset the Bell diagonal states, which were the only states for which an analytical expression of TDD had been worked out prior to our work. Our results are summarized in Table 1. Finally, we have illustrated the effectiveness of our findings in a specific paradigmatic problem where, despite being achievable, the analytical calculation of quantum discord is not informative. On the contrary, TDD is readily calculated in a simple X states † D (→) (ρ AB ) = 1 2 γ 2 1 max{γ 2 3 ,γ 2 2 +x 2 A3 }−γ 2 2 min{γ 2 3 ,γ 2 1 } max{γ 2 3 ,γ 2 2 +x 2 A3 }−min{γ 2 3 ,γ 2 1 }+γ 2 1 −γ 2 2 Table 1. Summary of the main results of the paper. We recall that the γ j 's indicate the (real) singular values of the correlation matrix Γ, with associated unit vectorsŵ j , while x A is the local Bloch vector of subsystem A, expressed in the coordinate system {ŵ j } 3 j=1 -see Section 2. * : In Section 3, the ordering |γ 1 | ≥ |γ 2 | ≥ |γ 3 | is assumed. # : We recall the standard form of a Quantum-Classical state: ρ AB = p ρ 0A ⊗ |0 B 0| + (1 − p)ρ 1A ⊗ |1 B 1|, where s j is the Bloch vector of ρ jA , s j = | s j | ( j = 0, 1) and ϕ is the smallest angle between s 0 and s 1 . † : In Section 5, |γ 1 | ≥ |γ 2 | is assumed, while no assumption is made on |γ 3 |. explicit form, being able at the same time to capture all the salient physical features of the QCs dynamics. Such approach could therefore prove particularly useful in order to clarify the role and physical meaning of QCs in a number of quantum coherent phenomena.
Due to the importance of quantum-classical and X states, along with the typical hindrances to the calculation of their QCs through bona fide measures, our work provides a significant contribution to the study of QCs quantifiers, by combining the desirable mathematical properties of TDD with its explicit computation for these classes of density matrices. Furthermore, we expect that the framework developed in this paper may be further exploited in future investigations to enlarge the class of quantum states that admit an analytical expression for TDD.